Dual applications of Chebyshev polynomials method: Efficiently finding thousands of central eigenvalues for many-spin systems

Computation of a large group of interior eigenvalues at the middle spectrum is an important problem for quantum many-body systems, where the level statistics provides characteristic signatures of quantum chaos. We propose an exact numerical method, dual applications of Chebyshev polynomials (DACP), to simultaneously find thousands of central eigenvalues, where the level space decreases exponentially with the system size. To disentangle the near-degenerate problem, we employ twice the Chebyshev polynomials, to construct an exponential semicircle filter as a preconditioning step and to generate a large set of proper basis states in the desired subspace. Numerical calculations on Ising spin chain and spin glass shards confirm the correctness and efficiency of DACP. As numerical results demonstrate, DACP is 30 times faster than the state-of-the-art shift-invert method for the Ising spin chain while 8 times faster for the spin glass shards. In contrast to the shift-invert method, the computation time of DACP is only weakly influenced by the required number of eigenvalues, which renders it a powerful tool for large scale eigenvalues computations. Moreover, the consumed memory also remains a small constant (5.6 GB) for spin-1/2 systems consisting of up to 20 spins, making it desirable for parallel computing.

Miocene and Pliocene Silicic Coromandel Volcanic Zone Tephras from ODP Site 1124-C: Petrogenetic Applications and Temporal Evolution

<p>The Coromandel Volcanic Zone (CVZ) was the longest-lived area of volcanism in New Zealand hosting the commencement of large explosive rhyolitic and ignimbrite forming eruptions. The NW trending Coromandel Peninsula is the subaerial remnant of the Miocene-Pliocene CVZ, which is regarded as a tectonic precursor to the Taupo Volcanic Zone (TVZ), currently the most dynamic and voluminous rhyolitic volcanic centre on Earth. This study presents new single glass shard major and trace element geochemical analyses for 72 high-silica volcanic tephra layers recovered from well-dated deep-sea sediments of the SW Pacific Ocean by the Ocean Drilling Program (ODP) Leg 181. ODP Site 1124, ~720 km south and east from the CVZ, penetrated sediments of the Rekohu Drift yielding an unprecedented record of major explosive volcanic eruptions owing to the favourable location and preservation characteristics at this site. This record extends onshore eruptive sequences of CVZ explosive volcanism that are obscured by poor exposure, alteration, and erosion and burial by younger volcanic deposits. Tephra layers recovered from Site 1124 are well-dated through a combination of biostratigraphic and palaeomagnetic methods allowing the temporal geochemical evolution of the CVZ to be reconstructed in relation to changes in the petrogenesis of CVZ arc magmas from ~ 10 to 2 Ma. This thesis establishes major and trace element geochemical "fingerprints" for all Site 1124-C tephras using well-established (wavelength dispersive electron probe microanalysis) and new (laser ablation inductively coupled plasma mass spectrometry) in situ single glass shard microanalytical techniques. Trace element analysis of Site 1124-C glass shards (as small as 20 um) demonstrate that trace element signatures offer a more specific, unequivocal characterisation for distinguishing (and potentially correlating) between tephras with nearly identical major element compositions. The Site 1124-C core contains 72 unaltered Miocene-Pliocene volcanic glass-shard-bearing laminae > 1 cm thick that correspond to 83 or 84 geochemical eruptive units. Revised eruptive frequencies based on the number of geochemical eruptive units identified represent at least one eruption every 99 kyr for the late Miocene and one per 74 kyr for the Pliocene. The frequency of tephra deposition throughout the history of the CVZ has not been constant, rather reflecting pulses of major explosive eruptions resulting in closely clustered groups of tephra separated by periods of reduced activity, relative volcanic quiescence or non-tephra deposition. As more regular activity became prevalent in the Pliocene, it was accompanied by more silicic magma compositions. Rhyolitic volcanic glass shards are characterised by predominantly calc-alkaline and minor high-K enriched major element compositions. Major element compositional variability of the tephras deposited between 10 Ma and 2 Ma reveals magma batches with pre-eruptive compositional gradients implying a broad control by fractional crystallisation. Trace element characterisation of glass shards reveals the role of magmatic processes that are not readily apparent in the relatively homogeneous major element compositions. Multi-element diagrams show prominent negative Sr and Ti anomalies against primitive mantle likely caused by various degrees of plagioclase and titanomagnetite fractional crystallisation in shallow magma chambers. Relative Nb depletion, characteristic of arc volcanism, is moderate in CVZ tephras. HFSEs (e.g. Nb, Zr, Ti) and HREEs (e.g. Yb, Lu) remain immobile during slab fluid flux suggesting they are derived from the mantle wedge. LILE (e.g. Rb, Cs, Ba, Sr) and LREE (e.g. La, Ce) enrichments are consistent with slab fluid contribution. B/La and Li/Y ratios can be used as a proxy for the flux of subducting material to the mantle wedge, they suggest there is a strong influence from this component in the generation of CVZ arc magmas, potentially inducing melting. CVZ tephra show long-term coherent variability in trace element geochemistry. Post ~ 4 Ma tephras display a more consistent, less variable, chemical fingerprint that persists up to and across the CVZ/TVZ transition at ~ 2 Ma. Initiation of TVZ volcanism may have occurred earlier than is presently considered, or CVZ to TVZ volcanism may have occurred without significant changes in magma generation processes.</p>

Miocene and Pliocene Silicic Coromandel Volcanic Zone Tephras from ODP Site 1124-C: Petrogenetic Applications and Temporal Evolution

<p>The Coromandel Volcanic Zone (CVZ) was the longest-lived area of volcanism in New Zealand hosting the commencement of large explosive rhyolitic and ignimbrite forming eruptions. The NW trending Coromandel Peninsula is the subaerial remnant of the Miocene-Pliocene CVZ, which is regarded as a tectonic precursor to the Taupo Volcanic Zone (TVZ), currently the most dynamic and voluminous rhyolitic volcanic centre on Earth. This study presents new single glass shard major and trace element geochemical analyses for 72 high-silica volcanic tephra layers recovered from well-dated deep-sea sediments of the SW Pacific Ocean by the Ocean Drilling Program (ODP) Leg 181. ODP Site 1124, ~720 km south and east from the CVZ, penetrated sediments of the Rekohu Drift yielding an unprecedented record of major explosive volcanic eruptions owing to the favourable location and preservation characteristics at this site. This record extends onshore eruptive sequences of CVZ explosive volcanism that are obscured by poor exposure, alteration, and erosion and burial by younger volcanic deposits. Tephra layers recovered from Site 1124 are well-dated through a combination of biostratigraphic and palaeomagnetic methods allowing the temporal geochemical evolution of the CVZ to be reconstructed in relation to changes in the petrogenesis of CVZ arc magmas from ~ 10 to 2 Ma. This thesis establishes major and trace element geochemical "fingerprints" for all Site 1124-C tephras using well-established (wavelength dispersive electron probe microanalysis) and new (laser ablation inductively coupled plasma mass spectrometry) in situ single glass shard microanalytical techniques. Trace element analysis of Site 1124-C glass shards (as small as 20 um) demonstrate that trace element signatures offer a more specific, unequivocal characterisation for distinguishing (and potentially correlating) between tephras with nearly identical major element compositions. The Site 1124-C core contains 72 unaltered Miocene-Pliocene volcanic glass-shard-bearing laminae > 1 cm thick that correspond to 83 or 84 geochemical eruptive units. Revised eruptive frequencies based on the number of geochemical eruptive units identified represent at least one eruption every 99 kyr for the late Miocene and one per 74 kyr for the Pliocene. The frequency of tephra deposition throughout the history of the CVZ has not been constant, rather reflecting pulses of major explosive eruptions resulting in closely clustered groups of tephra separated by periods of reduced activity, relative volcanic quiescence or non-tephra deposition. As more regular activity became prevalent in the Pliocene, it was accompanied by more silicic magma compositions. Rhyolitic volcanic glass shards are characterised by predominantly calc-alkaline and minor high-K enriched major element compositions. Major element compositional variability of the tephras deposited between 10 Ma and 2 Ma reveals magma batches with pre-eruptive compositional gradients implying a broad control by fractional crystallisation. Trace element characterisation of glass shards reveals the role of magmatic processes that are not readily apparent in the relatively homogeneous major element compositions. Multi-element diagrams show prominent negative Sr and Ti anomalies against primitive mantle likely caused by various degrees of plagioclase and titanomagnetite fractional crystallisation in shallow magma chambers. Relative Nb depletion, characteristic of arc volcanism, is moderate in CVZ tephras. HFSEs (e.g. Nb, Zr, Ti) and HREEs (e.g. Yb, Lu) remain immobile during slab fluid flux suggesting they are derived from the mantle wedge. LILE (e.g. Rb, Cs, Ba, Sr) and LREE (e.g. La, Ce) enrichments are consistent with slab fluid contribution. B/La and Li/Y ratios can be used as a proxy for the flux of subducting material to the mantle wedge, they suggest there is a strong influence from this component in the generation of CVZ arc magmas, potentially inducing melting. CVZ tephra show long-term coherent variability in trace element geochemistry. Post ~ 4 Ma tephras display a more consistent, less variable, chemical fingerprint that persists up to and across the CVZ/TVZ transition at ~ 2 Ma. Initiation of TVZ volcanism may have occurred earlier than is presently considered, or CVZ to TVZ volcanism may have occurred without significant changes in magma generation processes.</p>

A Study of Some Aspects of the Volcanic History of the Lake Taupo Area, North Island, New Zealand

<p>Rhyolitic pyroclastic eruptives from the Taupo area, New Zealand have been mapped as nine tephra formations of Holocene (0-10 kyr B.P.), and six of late Pleistocene age (20-c.50 kyr B.P.). Only the 10 younger tephras are dated by radiocarbon. All formations contain PLINIAN type airfall units but three, KAWAKAWA, WAIMIHIA and TAUPO also contain a major pyroclastic flow deposit (IGNIMBRIIE) unit. Dome extrusion can only be demonstrated for KARAPITI eruptive episode, but is inferred for the other Holocene episodes. TAUPO IGNIMBRITE is the product of the most recent eruption and is a particularly well preserved and extensive, unwelded pyroclastic flow deposit, up to 50m thick. Its variety of appearance is described in terms of three lithofacies; valley facies, fines depleted facies and veneer facies, each being formed by particular mechanisms within a pyroclastic flow. Abundant charred logs, lying prone within Taupo Ignimbrite, are radial about the source and attest to a radially outward moving mass dominated by laminar flow. Lake Taupo today covers most of the volcanic source area, preventing close examination and the identification of individual source vents. A vent for each Holocene tephra is inferred from isopachs, grainsize and lake bathymetry, but the vents so inferred show no spatial distribution with time. Nevertheless they are evenly spaced along a northeast trending line and lie on intersections with a northwest trending set of lineations, indicating deep, crustal, structural control on volcanism. Cumulative volume of airfall and ignimbrite material erupted in the Taupo area in the last 50 kyr has amounted to about 175 km3 of magma. Eruptions have proceeded in a step-wise manner, indicating the period to the next eruption is about 8 kyr. By the same approach, the next eruption from the Okataina area, 50 km to the north of Taupo is expected in less than 400 years. Whole rock and mineral chemistry clearly distinguishes between the Holocene and the late Pleistocene tephras, but within each group variations are subtle and no trends with time are apparent. None of the formations exhibit evidence for a chemically zoned magma body, but some data, especially pyroxene phenocryst chemistry, suggests magma inhomogeneities of mafic elements. The Holocene tephra were probably all erupted from the same magma chamber in which crystallisation was the dominant process but convection, crystal element diffusion and chamber replenishment were all probably operative. Results obtained by electron microprobe analysis of glass shards are critically dependent on the beam diameter and current used. By standardising these at 10 microns and 8 nanoamps respectively, comparable major element analyses on glass shards from numerous tephras ranging in age from 20 kyr to 600 kyr were obtained. The stratigraphic relationships between sets of samples (located mainly distal from source) and the close chemical similarity of some samples enabled a comprehensive tephrostratigraphy to be established. In particular, MT. CURL TEPHRA has a glass chemistry quite different from other stratigraphically separate tephras, establishing correlation of Mt. Curl Tephra to Whakamaru Ignimbrite. Likewise, other ignimbrite formations can be correlated to widespread airfall tephras, so establishing an absolute ignimbrite stratigraphy. Microprobe analysis of glass shards provides a method for indirectly determining the amount of hydration. For dated samples from a known weathering environment, the parameters controlling hydration can be quantified. For glass of uniform chemistry, shard size and porosity, ground temperature and groundwater movements are the most important parameters. No shards in the 63-250 micron size range have been found with more than 9% water, suggesting once this maximum is reached, glass rapidly alters to secondary products. Detailed knowledge of the volcanic history of the Taupo area, particularly since 50 kyrs B.P. allows the volcanic hazards of the region to be assessed. Fifteen major eruptions in 50 kyr gives a frequency of 1 in 3300 years, but the timing of individual events is not evenly spread throughout that time. Monitoring for volcanic Precursory events (not being undertaken at present) is essential to gauge the present and short-term future volcanic activity of the Taupo Volcanic Zone.</p>

A Study of Some Aspects of the Volcanic History of the Lake Taupo Area, North Island, New Zealand

<p>Rhyolitic pyroclastic eruptives from the Taupo area, New Zealand have been mapped as nine tephra formations of Holocene (0-10 kyr B.P.), and six of late Pleistocene age (20-c.50 kyr B.P.). Only the 10 younger tephras are dated by radiocarbon. All formations contain PLINIAN type airfall units but three, KAWAKAWA, WAIMIHIA and TAUPO also contain a major pyroclastic flow deposit (IGNIMBRIIE) unit. Dome extrusion can only be demonstrated for KARAPITI eruptive episode, but is inferred for the other Holocene episodes. TAUPO IGNIMBRITE is the product of the most recent eruption and is a particularly well preserved and extensive, unwelded pyroclastic flow deposit, up to 50m thick. Its variety of appearance is described in terms of three lithofacies; valley facies, fines depleted facies and veneer facies, each being formed by particular mechanisms within a pyroclastic flow. Abundant charred logs, lying prone within Taupo Ignimbrite, are radial about the source and attest to a radially outward moving mass dominated by laminar flow. Lake Taupo today covers most of the volcanic source area, preventing close examination and the identification of individual source vents. A vent for each Holocene tephra is inferred from isopachs, grainsize and lake bathymetry, but the vents so inferred show no spatial distribution with time. Nevertheless they are evenly spaced along a northeast trending line and lie on intersections with a northwest trending set of lineations, indicating deep, crustal, structural control on volcanism. Cumulative volume of airfall and ignimbrite material erupted in the Taupo area in the last 50 kyr has amounted to about 175 km3 of magma. Eruptions have proceeded in a step-wise manner, indicating the period to the next eruption is about 8 kyr. By the same approach, the next eruption from the Okataina area, 50 km to the north of Taupo is expected in less than 400 years. Whole rock and mineral chemistry clearly distinguishes between the Holocene and the late Pleistocene tephras, but within each group variations are subtle and no trends with time are apparent. None of the formations exhibit evidence for a chemically zoned magma body, but some data, especially pyroxene phenocryst chemistry, suggests magma inhomogeneities of mafic elements. The Holocene tephra were probably all erupted from the same magma chamber in which crystallisation was the dominant process but convection, crystal element diffusion and chamber replenishment were all probably operative. Results obtained by electron microprobe analysis of glass shards are critically dependent on the beam diameter and current used. By standardising these at 10 microns and 8 nanoamps respectively, comparable major element analyses on glass shards from numerous tephras ranging in age from 20 kyr to 600 kyr were obtained. The stratigraphic relationships between sets of samples (located mainly distal from source) and the close chemical similarity of some samples enabled a comprehensive tephrostratigraphy to be established. In particular, MT. CURL TEPHRA has a glass chemistry quite different from other stratigraphically separate tephras, establishing correlation of Mt. Curl Tephra to Whakamaru Ignimbrite. Likewise, other ignimbrite formations can be correlated to widespread airfall tephras, so establishing an absolute ignimbrite stratigraphy. Microprobe analysis of glass shards provides a method for indirectly determining the amount of hydration. For dated samples from a known weathering environment, the parameters controlling hydration can be quantified. For glass of uniform chemistry, shard size and porosity, ground temperature and groundwater movements are the most important parameters. No shards in the 63-250 micron size range have been found with more than 9% water, suggesting once this maximum is reached, glass rapidly alters to secondary products. Detailed knowledge of the volcanic history of the Taupo area, particularly since 50 kyrs B.P. allows the volcanic hazards of the region to be assessed. Fifteen major eruptions in 50 kyr gives a frequency of 1 in 3300 years, but the timing of individual events is not evenly spread throughout that time. Monitoring for volcanic Precursory events (not being undertaken at present) is essential to gauge the present and short-term future volcanic activity of the Taupo Volcanic Zone.</p>

TephraNZ: a major- and trace-element reference dataset for glass-shard analyses from prominent Quaternary rhyolitic tephras in New Zealand and implications for correlation

Although analyses of tephra-derived glass shards have been undertaken in New Zealand for nearly four decades (pioneered by Paul Froggatt), our study is the first to systematically develop a formal, comprehensive, open-access reference dataset of glass-shard compositions for New Zealand tephras. These data will provide an important reference tool for future studies to identify and correlate tephra deposits and for associated petrological and magma-related studies within New Zealand and beyond. Here we present the foundation dataset for TephraNZ, an open-access reference dataset for selected tephra deposits in New Zealand. Prominent, rhyolitic, tephra deposits from the Quaternary were identified, with sample collection targeting original type sites or reference locations where the tephra's identification is unequivocally known based on independent dating and/or mineralogical techniques. Glass shards were extracted from the tephra deposits, and major- and trace-element geochemical compositions were determined. We discuss in detail the data reduction process used to obtain the results and propose that future studies follow a similar protocol in order to gain comparable data. The dataset contains analyses of glass shards from 23 proximal and 27 distal tephra samples characterising 45 eruptive episodes ranging from Kaharoa (636 ± 12 cal yr BP) to the Hikuroa Pumice member (2.0 ± 0.6 Ma) from six or more caldera sources, most from the central Taupō Volcanic Zone. We report 1385 major-element analyses obtained by electron microprobe (EMPA), and 590 trace-element analyses obtained by laser ablation (LA)-ICP-MS, on individual glass shards. Using principal component analysis (PCA), Euclidean similarity coefficients, and geochemical investigation, we show that chemical compositions of glass shards from individual eruptions are commonly distinguished by major elements, especially CaO, TiO2, K2O, and FeOtt (Na2O+K2O and SiO2/K2O), but not always. For those tephras with similar glass major-element signatures, some can be distinguished using trace elements (e.g. HFSEs: Zr, Hf, Nb; LILE: Ba, Rb; REE: Eu, Tm, Dy, Y, Tb, Gd, Er, Ho, Yb, Sm) and trace-element ratios (e.g. LILE/HFSE: Ba/Th, Ba/Zr, Rb/Zr; HFSE/HREE: Zr/Y, Zr/Yb, Hf/Y; LREE/HREE: La/Yb, Ce/Yb). Geochemistry alone cannot be used to distinguish between glass shards from the following tephra groups: Taupō (Unit Y in the post-Ōruanui eruption sequence of Taupō volcano) and Waimihia (Unit S); Poronui (Unit C) and Karapiti (Unit B); Rotorua and Rerewhakaaitu; and Kawakawa/Ōruanui, and Okaia. Other characteristics, including stratigraphic relationships and age, can be used to separate and distinguish all of these otherwise-similar tephra deposits except Poronui and Karapiti. Bimodality caused by K2O variability is newly identified in Poihipi and Tahuna tephras. Using glass-shard compositions, tephra sourced from Taupō Volcanic Centre (TVC) and Mangakino Volcanic Centre (MgVC) can be separated using bivariate plots of SiO2/K2O vs. Na2O+K2O. Glass shards from tephras derived from Kapenga Volcanic Centre, Rotorua Volcanic Centre, and Whakamaru Volcanic Centre have similar major- and trace-element chemical compositions to those from the MgVC, but they can overlap with glass analyses from tephras from Taupō and Okataina volcanic centres. Specific trace elements and trace-element ratios have lower variability than the heterogeneous major-element and bimodal signatures, making them easier to fingerprint geochemically.

TephraNZ: a major and trace element reference dataset for prominent Quaternary rhyolitic tephras in New Zealand and implications for correlation

Although analyses of tephra-derived glass shards have been undertaken in New Zealand for nearly four decades (pioneered by Paul Froggatt), our study is the first to systematically develop a formal, comprehensive, open access, reference dataset of glass-shard compositions for New Zealand tephras. These data will provide an important reference tool for future studies to identify and correlate tephra deposits and for associated petrological and magma-related studies within New Zealand and beyond. Here we present the foundation dataset for TephraNZ, an open access reference dataset for selected tephra deposits in New Zealand. Prominent, rhyolitic, tephra deposits from the Quaternary were identified, with sample collection targeting original type sites or reference locations where the tephra's identification is unequivocally known based on independent dating or mineralogical techniques. Glass shards were extracted from the tephra deposits and major and trace element geochemical compositions were determined. We discuss in detail the data reduction process used to obtain the results and propose that future studies follow a similar protocol in order to gain comparable data. The dataset contains analyses of twenty-three proximal and twenty-seven distal tephra samples characterising 45 eruptive episodes ranging from Kaharoa (636 &amp;pm; 12 cal. yrs BP) to the Hikuroa Pumice member (2.0 &amp;pm; 0.6 Ma) from six or more caldera sources, most from the central Taupō Volcanic Zone. We report 1385 major element analyses obtained by electron microprobe (EMPA), and 590 trace element analyses obtained by laser ablation (LA)-ICP-MS, on individual glass shards. Using PCA, Euclidean similarity coefficients, and geochemical investigation, we show that chemical compositions of glass shards from individual eruptions are commonly distinguished by major elements, especially CaO, TiO2, K2O, FeOt (Na2O+ K2O and SiO2/K2O), but not always. For those tephras with similar glass major-element signatures, some can be distinguished using trace elements (e.g. HFSEs: Zr, Hf, Nb; LILE: Ba, Rb; REE: Eu, Tm, Dy, Y, Tb, Gd, Er, Ho, Yb, Sm), and trace element ratios (e.g. LILE / HFSE: Ba / Th, Ba / Zr, Rb / Zr; HFSE / HREE: Zr / Y, Zr / Yb, Hf / Y; LREE / HREE: La / Yb, Ce / Yb). Geochemistry alone cannot be used to distinguish between glass shards from the following tephra groups: Taupō (Unit Y in the post-Ōruanui eruption sequence of Taupō volcano) and Waimihia (Unit S); Poronui (Unit C) and Karapiti (Unit B); Rotorua and Rerewhakaaitu; and Kawakawa/Ōruanui, Okaia, and Unit L (of the Mangaone subgroup eruption sequence). Other characteristics can be used to separate and distinguish all of these otherwise-similar eruptives except Poronui and Karapiti. Bimodality caused by K2O variability is newly identified in Poihipi and Tahuna tephras. Using glass shard compositions, tephra sourced from Taupō Volcanic Centre (TVC) and Mangakino Volcanic Centre (MgVC) can be separated using bivariate plots of SiO2/K2O vs. Na2O+K2O. Glass shards from tephras derived from Kapenga Volcanic Centre, Rotorua Volcanic Centre, and Whakamaru Volcanic Centre have similar major- and trace-element chemical compositions to those from the MgVC, but can overlap with glass analyses from tephras from Taupō and Okataina volcanic centres. Specific trace elements and trace element ratios have lower variability than the heterogeneous major element and bimodal signatures, making them easier to geochemically fingerprint.